JAMES BERNOULLI. 71 



c f % 



1 — , 1 — ^, 1 — • Then the resulting probability of the con- 



elusion is 1 — ~- . This is obvious from the consideration that 

 adg 



any one of the arguments would establish the conclusion, so that 



the conclusion fails only when all the arguments fail. 



Supj)ose now that we have in addition two arguments of the 



mixed kind : let their respective probabilities be — ^^ , . 



Then James Bernoulli gives for the resulting probability 



, cfiru 



1 — -^ 



adg (ru + qt) ' 



But this formula is inaccurate. For the supposition q = amounts 

 to having one argument absolutehj decisive against the conclusion, 

 while yet the formula leaves still a certain probability for the 

 conclusion. The error was pointed out by Lambert; see Pre vest 

 and Lhuilier, Memoir es de F Acad.... Berliii iov 1797. 



123. The most remarkable subject contained in the fourth 

 part of the Ars Conjectandi is the enunciation and investigation 

 of what we now call Bernoulli s Theorem. It is introduced in 

 terms which shew a high opinion of its importance : 



Hoc igitur est illud Problema, quod evulgauduni hoc loco proposui, 

 postquam jam per vicenniiini pressi, et cujus turn novitas, turn summa 

 utilitas cum pari conjuucta difficultate omnibus reliquis hujus doc- 

 triiiae capitibus pondus et pretium superaddere potest. Ars Conjectandij 

 page 227. See also De Moivre's Doctrine of Chances , page 2d^. 



We will now state the purely algebraical part of the theorem. 

 Suppose that (r + s)**' is exj)anded by the Binomial Theorem, the 

 letters all denoting integral numbers and t being equal to r + s. 

 Let u denote the sum of the greatest term and the n preceding 

 terms and the n following terms. Then by taking n large enough 

 the ratio of u to the sum of all the remaining terms of the expan- 

 sion may be made as gi-eat as we please. 



If we wish that this ratio should not be less than c it will be 

 sufficient to take n equal to the greater of the two following ex- 

 pressions, 



